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Chapter 2: Problem 147

Translate to an equation and then solve. \(r\) divided by 12 is equal to -48.

Short Answer

Expert verified
r = -576

Step by step solution

01

- Translate the Sentence to an Equation

The sentence says that the variable \( r \) divided by 12 is equal to -48. This can be translated to the equation: \[ \frac{r}{12} = -48 \]
02

- Multiply Both Sides by 12

To solve for \( r \), multiply both sides of the equation by 12 to isolate the variable \( r \): \[ 12 \times \frac{r}{12} = 12 \times (-48) \]
03

- Simplify the Equation

Simplify both sides of the equation: \[ r = -576 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Translating Sentences to Equations
When given a word problem or a sentence describing a mathematical relationship, the first step is to translate that sentence into an equation. In our example, the sentence says that a variable, we'll call it \( r \), divided by 12, equals -48. This can be directly translated into the mathematical expression:
\[ \frac{r}{12} = -48 \]
Here, the fraction \( \frac{r}{12} \) represents 'r divided by 12'. The equation then states that this fraction is equal to -48.

Translating sentences to equations helps in visualizing and systematically solving the problem. Keep in mind common words used in problems and their mathematical counterparts:
  • 'Sum/Add' translates to \(+\).
  • 'Difference/Subtract' translates to \(-\).
  • 'Product/Times' translates to \(\times\).
  • 'Quotient/Divided by' translates to a fraction.
Isolation of Variables
Once you have translated a sentence into an equation, the next step is to solve for the unknown variable. This process involves isolating the variable on one side of the equation.
In our case, the equation is:
\[ \frac{r}{12} = -48 \]
We want to solve for \( r \). To do this, we need \( r \) to be by itself on one side of the equation. This can be done using various algebraic operations like adding, subtracting, multiplying, or dividing both sides of the equation by the same number.

For the above equation, we need to get rid of the 12 in the denominator of the fraction. This leads us to the next important step.
Multiplication Property of Equality
The multiplication property of equality states that if you multiply both sides of an equation by the same number, your equation remains balanced.

Let's apply this property to our sample equation \( \frac{r}{12} = -48 \). To isolate \( r \), we multiply both sides of the equation by 12:
\[ 12 \times \frac{r}{12} = 12 \times (-48) \]
On the left side, the 12s cancel out, leaving us with:

\[ r = 12 \times -48\]

Now, simplify the right side:

\[ r = -576 \]

So, the solution to the equation is \( r = -576 \).

This shows that using the multiplication property of equality can effectively isolate the variable you're solving for.

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Most popular questions from this chapter

Elevators The total weight, \(w\), of an elevator's passengers can be no more than 1,200 pounds. Write this as an inequality.

Use the formula \(d=r t\). Solve for \(r\) (a) when \(d=160\) and \(t=2.5\) (b) in general

Solve each inequality, graph the solution on the number line, and write the solution in interval notation. $$\frac{n}{13} \leq-6$$

Explain how you find the least common denominator of \(\frac{3}{8}, \frac{1}{6},\) and \(\frac{2}{3}\).

Solve each inequality, graph the solution on the number line, and write the solution in interval notation. $$b+\frac{7}{8} \geq \frac{1}{6}$$
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